Question: $ 0.\overline{3} \div -1.\overline{79} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 3.3333...\\ x &= 0.3333...\end{align*} $ $\begin{align*} 9x &= 3 \\ x &= \dfrac{3}{9}\end{align*} $ $\begin{align*} 100y &= -179.798...\\ y &= -1.798...\end{align*} $ $\begin{align*} 99y &= -178 \\ y &= -\dfrac{178}{99}\end{align*} $ So, the problem becomes: $ \dfrac{3}{9} \div -\dfrac{178}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{3}{9} \times -\dfrac{99}{178} = {?} $ $ \phantom{\dfrac{3}{9} \times -\dfrac{178}{99}} = \dfrac{3 \times 99}{9 \times -178} $ $ \phantom{\dfrac{3}{9} \times -\dfrac{178}{99}} = \dfrac{3 \times \cancel{99}11} {\cancel{9} \times -178} $ $ \phantom{\dfrac{3}{9} \times -\dfrac{178}{99}} = -\dfrac{33}{178} $